7.3    Extensions to a General Class of Discrete-Time Nonlinear Systems

In this section, we extend the results of the previous sections to a more general class of discrete-time nonlinear systems that may not necessarily be affine in u and w. We consider the general class of systems described by the state-space equations on $\mathcal{X}\subset {\Re }^n$ containing the origin {0}:

${\sum }^d:\{\begin{array}{l}{x}_{k+1}=F(x_k,w_k,u_k);x(k_0)=x^0\\ z_k=Z(x_k,w_k,u_k)\\ y_k=Y(x_k,w_k)\end{array}$

(7.64)

where all the variables have their usual meanings, and $F:\mathcal{X}\times \mathcal{W}\times \mathcal{U}\to \mathcal{X},F:\left(0,0,0\right)=0,Z:\mathcal{X}\times \mathcal{W}\times \mathcal{U}\to {\Re }^s,Z\left(0,0,0\right)=0,Y:\mathcal{X}\times \mathcal{W}\to {\Re }^m$. We begin with the full-information and state-feedback problems.

7.3.1    Full-Information ${\mathcal{H}}_{\infty }$-Control for a General Class of Discrete-Time Nonlinear Systems

To solve the full-information and state-feedback problems for the system Σ^d (7.64), we consider the Hamiltonian function:

$\tilde{H}(x,u,w)=\tilde{V}(F(x,w,u))-\tilde{V}(x)+\frac{1}{2}({\Vert Z(x,u,w)\Vert }^2-{\text{γ}}^2{\Vert w\Vert }^2)$

(7.65)

for some positive-definite function $\tilde{V}:\mathcal{X}\to {\Re }_{+}$, and let

$\frac{{\partial }^2\tilde{H}}{\partial {(u,w)}^2}(0,0,0)=\left[\begin{array}{cc}\frac{{\partial }^2\tilde{H}}{\partial u^2}& \frac{{\partial }^2\tilde{H}}{\partial w\partial u}\\ \frac{{\partial }^2\tilde{H}}{\partial u\partial w}& \frac{{\partial }^2\tilde{H}}{\partial w^2}\end{array}\right](0,0,0)$

where

$\begin{array}{l}{h}_{uu}(0)={\left[{\left(\frac{\partial F}{\partial u}\right)}^T\frac{{\partial }^2\tilde{V}}{\partial {\text{λ}}^2}(0)\left(\frac{\partial F}{\partial u}\right)+{\left(\frac{\partial Z}{\partial u}\right)}^T\left(\frac{\partial Z}{\partial u}\right)\right]|}_{x=0,u=0,w=0}\\ h_{ww}(0)={\left[{\left(\frac{\partial F}{\partial w}\right)}^T\frac{{\partial }^2\tilde{V}}{\partial {\text{λ}}^2}(0)\left(\frac{\partial F}{\partial w}\right)+{\left(\frac{\partial Z}{\partial w}\right)}^T\left(\frac{\partial Z}{\partial w}\right)-{\text{γ}}^{2}I\right]|}_{x=0,u=0,w=0}\\ h_{uw}(0)={\left[{\left(\frac{\partial F}{\partial u}\right)}^T\frac{{\partial }^2\tilde{V}}{\partial {\text{λ}}^2}(0)\left(\frac{\partial F}{\partial w}\right)+{\left(\frac{\partial Z}{\partial u}\right)}^T\left(\frac{\partial Z}{\partial w}\right)\right]|}_{x=0,u=0,w=0}\end{array}$

Suppose now that the following assumption holds:

(GN1)

$h_{uu}(0)>0,h_{ww}(0)-h_{wu}(0)h_{uu}^{-1}(0)h_{uw}(0)<0$

Then by the Implicit-function Theorem, the above assumption implies that there exist unique smooth functions ${\tilde{u}}^{\star }\left(x\right),{\tilde{w}}^{\star }\left(x\right)$ with ũ^⋆ (0) = 0 and ${\tilde{w}}^{\star }\left(0\right)=0$ defined in a neighborhood $\overline{X}$

of x = 0 and satisfying the functional equations

$\begin{array}{l}0=\frac{\partial \tilde{H}}{\partial u}(x,{\tilde{u}}^{\star }(x),{\tilde{w}}^{\star }(x))\hfill \\ =(\frac{\partial \tilde{V}}{\partial F}\frac{\partial F}{\partial u}+Z^T\frac{\partial Z}{\partial u}){|}_{u={\tilde{u}}^{\star }(x),w={\tilde{w}}^{\star }(x)}\hfill \end{array}$

(7.66)

$\begin{array}{l}0=\frac{\partial \tilde{H}}{\partial w}(x,{\tilde{u}}^{\star }(x),{\tilde{w}}^{\star }(x))\\ ={\left(\frac{\partial \tilde{V}}{\partial F}\frac{\partial F}{\partial w}+Z^T\frac{\partial Z}{\partial w}\right)|}_{u={\tilde{u}}^{\star }(x),w={\tilde{w}}^{\star }(x).}\end{array}$

(7.67)

Similarly, let

$\begin{array}{l}{h}_{uu}(x)\triangleq \frac{{\partial }^2\tilde{H}}{\partial u^2}(x,{\tilde{u}}^{\star }(x),{\tilde{w}}^{\star }(x))\\ h_{ww}(x)\triangleq \frac{{\partial }^2\tilde{H}}{\partial w^2}(x,{\tilde{u}}^{\star }(x),{\tilde{w}}^{\star }(x))\\ h_{uw}(x)\triangleq \frac{{\partial }^2\tilde{H}}{\partial w\partial u}(x,{\tilde{u}}^{\star }(x),{\tilde{w}}^{\star }(x))=h_{wu}^T(x,{\tilde{u}}^{\star },{\tilde{w}}^{\star })\end{array}$

be associated with the optimal solutions ${\tilde{u}}^{\star }\left(x\right),{\tilde{w}}^{\star }\left(x\right)$, and let the corresponding DHJIE associated with (7.64) be denoted by

(GN2)

$\tilde{V}(F(x,{\tilde{w}}^{\star }(x),{\tilde{u}}^{\star }(x)))-\tilde{V}(x)+\frac{1}{2}({\Vert Z(x,{\tilde{u}}^{\star }(x),{\tilde{w}}^{\star }(x))\Vert }^2-{\text{γ}}^2{\Vert {\tilde{w}}^{\star }(x)\Vert }^2)=0,\tilde{V}(0)=0.$

(7.68)

Then we have the following result for the solution of the full-information problem.

Theorem 7.3.1 Consider the discrete-time nonlinear system (7.64), and suppose there exists a C² positive-definite function $\tilde{V}:X_1\subset \mathcal{X}\to {\Re }_{+}$ locally defined in a neighborhood X₁ of x = 0 satisfying the hypotheses (GN1), (GN2). In addition, suppose the following assumption is also satisfied by the system:

(GN3) Any bounded trajectory of the free system

$x_{k+1}=F(x_k,0,u_k),$

under the constraint

$Z(x_k,0,u_k)=0$

for all k ∈ Z₊, is such that, lim_k→∞ x_k = 0.

Then there exists a static full-information feedback control

$u_k={\overline{u}}^{\star }(x_k)-h_{uu}^{-1}(x_k)h_{uw}(x_k)(w_k-{\overline{w}}^{\star }(x_k))$

which solves the DFIFBNLHICP for the system.

Proof: The proof can be pursued along similar lines as Theorem 7.1.2. □

The above result can also be easily specialized to the state-feedback case as follows.

Theorem 7.3.2 Consider the discrete-time nonlinear system (7.64), and suppose there exists a C² positive-definite function $\tilde{V}:X_2\subset \mathcal{X}\to {\Re }_{+}$ locally defined in a neighborhood X₂ of x = 0 satisfying the hypothesis

(GN1s)

$h_{uu}(0)>0,h_{ww}(0)<0$

and hypotheses (GN2), (GN3) above. Then, the static state-feedback control

$u_k={\overline{u}}^{\star }(x_k)$

solves the DSFBNLHICP for the system.

Proof: The theorem can be proven along similar lines as Theorem 7.1.3.□

Moreover, the parametrization of all static state-feedback controllers can also be given in the following theorem.

Theorem 7.3.3 Consider the discrete-time nonlinear system (7.64), and suppose the following hypothesis holds

(GN2s) there exists a C² positive-definite function $\tilde{V}:X_3\subset \mathcal{X}\to {\Re }_{+}$ locally defined in a neighborhood X₃ of x = 0 satisfying the DHJIE

$\begin{array}{l}\tilde{V}(F(x,{\tilde{w}}^{\star }(x),{\tilde{u}}^{\star }(x)))-\tilde{V}(x)+\frac{1}{2}({\Vert Z(x,{\tilde{u}}^{\star }(x),{\tilde{w}}^{\star }(x))\Vert }^2-{\text{γ}}^2{\Vert {\tilde{w}}^{\star }(x)\Vert }^2)\\ =-\psi (x)[h_{uu}(x)-h_{uw}(x)h_{ww}^{-1}(x)h_{21}(x)\psi (x),\tilde{V}(0)=0\end{array}$

(7.69)

for some arbitrary smooth function ψ : X₃ → ℜ^p, ψ(0) = 0,

as well as the hypotheses (GN3) and (GN1s) with $\tilde{V}$ in place of $\tilde{V}$. Then, the family of controllers

${\text{K}}_{SFg}=\left\{u_k|u_k={\tilde{u}}^{\star }(x_k)+\psi (x_k)\right\}$

(7.70)

is a parametrization of all static state-feedback controllers that solves the DSFBNLHICP for the system.

7.3.2    Output Measurement-Feedback ${\mathcal{H}}_{\infty }$-Control for a General Class of Discrete-Time Nonlinear Systems

In this subsection, we discuss briefly the output measurement-feedback problem for the general class of nonlinear systems (7.64). Theorem 7.2.1 can easily be generalized to this class of systems. As in the previous case, we can postulate the existence of a dynamic compensator of the form:

${\tilde{\sum }}_{dynobs}^{dc}:\{\begin{array}{l}{\xi }_{k+1}=F({\xi }_k,{\tilde{w}}^{\star }({\xi }_k),{\tilde{u}}^{\star }({\xi }_k))+\tilde{G}({\xi }_k)(y_k-Y({\xi }_k,{\tilde{w}}^{\star }({\xi }_k))\\ u_k={\tilde{u}}^{\star }({\xi }_k)\end{array}$

(7.71)

where $\tilde{G}$(.) is the output-injection gain matrix, and ${\tilde{w}}^{\star }(.),{\tilde{u}}^{\star }(.)$, are the solutions to equations (7.66), (7.67). Let the closed-loop system (7.64) with the controller (7.71) be represented as

$\begin{array}{l}{x}_{k+1}^e=F^e(x_k^e,w_k)\\ z_k=Z^e(x_k^e,w_k)\end{array}$

where $x^e={\left[\begin{array}{cc}{x}^T& {\xi }^T\end{array}\right]}^T,$

$F^e(x^e,w)=\left[\begin{array}{l}F(x,w,{\tilde{u}}^{\star })\\ F(x,{\tilde{w}}^{\star }(\xi ),{\tilde{u}}^{\star }(\xi ))+\tilde{G}(\xi )(y(x,w)-y(\xi ,{\tilde{w}}^{\star }(\xi )))\end{array}\right],$

$Z^e(x^e,w)=Z(x,w,{\tilde{u}}^{\star }(\xi )).$

Then the following result is a direct extension of Theorem 7.2.1.

Theorem 7.3.4 Consider the discrete-time nonlinear system (7.64) and assume the following:

(i)  Assumption (GN3) holds and rank $\kappa \left\{\frac{\partial Z}{\partial u}\left(0,0,0\right)\right\}=p$.

(ii)  There exists a C² positive-definite function $\tilde{V}:X\to {\Re }_{+}$locally defined in a neighborhood X of x = 0 satisfying Assumption (GN2).

(iii)  There exists an output-injection gain matrix $\tilde{G}(.)$ and a C² real-valued function W : X₄ × X₄ locally defined in a neighborhood X₄ × X of (x, ξ) = (0, 0), X₄ ∩ X₄ ≠ ∅, with W (0, 0) = 0, W (x, ξ) > 0 ∀x ≠ ξ and satisfying

(GNM1)

$F^{e^T}(0,0,0)\frac{{\partial }^{2}W}{\partial x^{e^2}}(0,0)F^e(0,0,0)+h_{ww}(0)<0;$

(GNM2)

$\begin{array}{l}W(F^e(x^e,{\tilde{\alpha }}_1(x^e)-W(x^e)+V(F(x,{\tilde{\alpha }}_1(x^e)u^{\star }(\xi )))-V(x)+\\ \frac{1}{2}({\Vert Z(x,{\tilde{\alpha }}_1(x^e),{\tilde{u}}^{\star }(\xi ))\Vert }^2-{\text{γ}}^2{\Vert {\tilde{\alpha }}_1(x^e)\Vert }^2)=0,\end{array}$

where ${\tilde{\alpha }}_1\left(x^e\right)=0$ with ${\tilde{\alpha }}_1\left(0\right)=0$is a locally unique solution of the equation

$\begin{array}{l}{\frac{\partial W}{\partial \beta }|}_{\beta =F^e(x^e,w)}\frac{\partial F^e}{\partial w}(x^e,w)+{\frac{\partial V}{\partial \text{λ}}|}_{\text{λ}=F(x,w,\tilde{u}\star (\xi ))}\frac{\partial F}{\partial w}(x,w,{\tilde{u}}^{\star }(\xi ))+\\ Z^T(x,,w,{\tilde{u}}^{\star }(\xi ))\frac{\partial Z}{\partial w}(x,w,{\tilde{u}}^{\star }(\xi ))-{\text{γ}}^2{w}^T=0\end{array}$

(GNM3) The discrete-time nonlinear system

$x_{\kappa +1}=F(\xi ,{\tilde{w}}^{\star }(\xi ),0)-\tilde{G}(\xi )Y(\xi ,{\tilde{w}}^{\star }(\xi ))$

is locally asymptotically-stable at ξ = 0.

Then, the DMFBNLHICP for the system (7.64) is solvable with the compensator (7.71).

7.4    Approximate Approach to the Discrete-Time Nonlinear ${\mathcal{H}}_{\infty }$-Control Problem

In this section, we discuss alternative approaches to the discrete-time nonlinear ${\mathcal{H}}_{\infty }$-Control problem for affine systems. It should have been observed in Sections 7.1, 7.2, that the control laws that were derived are given implicitly in terms of solutions to certain pairs of algebraic equations. This makes the computational burden in using this design method more intensive. Therefore, in this section, we discuss alternative approaches, although approximate, but which can yield explicit solutions to the problem. We begin with the state-feedback problem.

7.4.1    An Approximate Approach to the Discrete-Time State-Feedback Problem

We consider again the nonlinear system (7.1), and assume the following.

Assumption 7.4.1

$rank\left\{k_{12}\left(x\right)\right\}=p.$

Reconsider now the Hamiltonian function (7.11) associated with the problem:

$\begin{array}{l}{H}_2(w,u)=V(f(x)+g_1(x)w+g_2(x)u)-V(x)+\frac{1}{2}({\Vert h_1(x)+k_{11}(x)w+k_{12}(x)u\Vert }^2-\\ {\text{γ}}^2{\Vert w\Vert }^2)\end{array}$

(7.72)

for some smooth positive-definite function V : $\mathcal{X}$ → ℜ₊. Suppose there exists a smooth real-valued function $\overline{u}(x)\in {\Re }^p,\overline{u}(0)=0$ such that the HJI-inequality

$H_2(w,\overline{u}(x))<0$

(7.73)

is satisfied for all x ∈ $\mathcal{X}$ and w ∈ $\mathcal{W}$. Then it is clear from the foregoing that the control law

$u=\overline{u}\left(x\right)$

solves the DSFBNLHICP for the system Σ^da globally. The bottleneck however, is in getting an explicit form for the function ū(x). This problem stems from the first term in the HJI-inequality (7.73), i.e.,

$V(f(x)+g_1(x)w+g_2(x)u,$

which is a composition of functions and is not necessarily quadratic in u, as in the continuous-time case. Thus, suppose we replace this term by an approximation which is “quadratic” in (w, u), and nonlinear in x, i.e.,

$V(f(x)+\upsilon )=V(f(x))+V_x(f(x))\upsilon +\frac{1}{2}{\upsilon }^T{V}_{xx}(f(x))\upsilon +R_m(x,\upsilon )$

for some vector function v ∈ $\mathcal{X}$ and where R_m is a remainder term such that

$\underset{\upsilon \to 0}{\lim }\frac{R_m(x,\upsilon )}{{\Vert \upsilon \Vert }^2}=0$

Then, we can seek a saddle-point for the new Hamiltonian function

$\begin{array}{l}{\widehat{H}}_2(w,u)=V(f(x))+V_x(f(x))(g_1(x)w+g_2(x)u)+\\ \frac{1}{2}{\left(g_1(x)w+g_2(x)u\right)}^T{V}_{xx}(f(x))(g_1(x)+g_2(x)u)-V(x)+\\ \frac{1}{2}{\Vert h_1(x)+k_{11}(x)w+k_{12}(x)u\Vert }^2-\frac{1}{2}{\text{γ}}^2{\Vert w\Vert }^2\end{array}$

(7.74)

by neglecting the higher-order term R_m(x, g₁(x)w + g₂(x)u). Since ${\widehat{H}}_2$(u, w) is quadratic in (w, u), it can be represented as

${\widehat{H}}_2(w,u)=V(f(x))-V(x)+\frac{1}{2}{h}_1^T(x)h_1(x)+\widehat{S}(x)\left[\begin{array}{l}w\\ u\end{array}\right]+\frac{1}{2}\left[\begin{array}{l}w\\ u\end{array}\right]\widehat{R}(x)\left[\begin{array}{l}w\\ u\end{array}\right],$

where

$\widehat{S}(x)=h_1^T(x)\left[k_{11}(x)k_{12}(x)\right]+V_x(f(x))\left[g_1(x)g_2(x)\right]$

and

$\widehat{R}(x)=\left(\begin{array}{l}{k}_{11}^T(x)k_{11}(x)-{\text{γ}}^{2}I{k}_{11}^T(x)k_{12}(x)\\ k_{12}^T(x)k_{11}(x)k_{12}^T(x)k_{12}(x)\end{array}\right)+\left(\begin{array}{l}{g}_1^T(x)\\ g_2^T(x)\end{array}\right)V_{xx}(f(x))(g_1(x)g_2(x))$

From this, it is easy to determine conditions for the existence of a unique saddle-point and explicit formulas for the coordinates of this point. It can immediately be determined that, if $\widehat{R}\left(x\right)$ is nonsingular, then

${\widehat{H}}_2(w,u)={\widehat{H}}_2(w^{\star }(x),u^{\star }(x))+\frac{1}{2}{\left[\begin{array}{l}w-{\widehat{w}}^{\star }(x)\\ u-{\widehat{u}}^{\star }(x)\end{array}\right]}^T\widehat{R}(x)\left[\begin{array}{l}w-{\widehat{w}}^{\star }(x)\\ u-{\widehat{u}}^{\star }(x)\end{array}\right]$

(7.75)

where

$\left[\begin{array}{l}{\widehat{w}}^{\star }(x)\\ {\widehat{u}}^{\star }(x)\end{array}\right]=-{\widehat{R}}^{-1}(x){\widehat{S}}^T(x).$

(7.76)

One condition that gurantees that $\widehat{R}\left(x\right)$ is nonsingular (by Assumption 7.4.1) is that the submatrix

${\widehat{R}}_{11}(x):=k_{11}{}^{T_{{}_{}}}(x)k11(x)-{\gamma }^{2}I+1/2g_1{}^T(x)Vxx(f(x))g_1(x)<0\forall \mathrm{x.}$

If the above condition is satisfied for some γ > 0, then ${\widehat{H}}_2\left(x,w,u\right)$ has a saddle-point at (û, ŵ ), and

${\widehat{H}}_2(\widehat{w}{}^{\star }(x),\widehat{u}{}^{\star }(x))=V(f(x))-V(x)-\frac{1}{2}\widehat{S}(x){\widehat{R}}^{-1}(x){\widehat{S}}^T(x)+\frac{1}{2}{h}_1^T(x)h_1(x).$

(7.77)

The above development can now be summarized in the following lemma.

Lemma 7.4.1 Consider the discrete-time nonlinear system (7.1), and suppose there exists a smooth positive-definite function V : X₀ ⊂ $\mathcal{X}$ → ₊, V (0) = 0 and a positive number δ > 0 such that

(i)  

${\widehat{H}}_2(\widehat{w}\star (x),\widehat{u}\star (x))<0\forall 0\ne x\in X_0$

(7.78)

(ii)  

${\widehat{H}}_2(\widehat{w}\star (x),\widehat{u}\star (x))<-\frac{1}{2}\text{δ(||}\widehat{w}\star (x)|{|}^2+||\widehat{u}\star (x)|{|}^2)\forall x\in X_0,$

(iii)  

${\widehat{R}}_{11}(x)<0\forall x\in X_0.$

Then, there exists a neighborhood X × W of (w, x) = (0, 0) in $\mathcal{X}\times \mathcal{W}$ such that V satisfies the HJI-inequality (7.73) with ū = û^⋆ (x), û^⋆ (0) = 0.

Proof: By construction, H₂(x, w, u) satisfies

${\widehat{H}}_2(x,w,u)={\widehat{H}}_2({\widehat{w}}^{\star }(x),{\widehat{u}}^{\star }(x))+\frac{1}{2}{\left(w-{\widehat{w}}^{\star }\right)}^T{\widehat{R}}_{11}(x)(w-{\widehat{w}}^{\star }(x)).$

Since R₁₁(0) is negative-definite by hypothesis (iii), there exists a neighborhood X₁ of x = 0 and a positive number c > 0 such that

${\left(w-{\widehat{w}}^{\star }\right)}^T{\widehat{R}}_{11}(x)(w-{\widehat{w}}^{\star }(x))\le c{\Vert (w-{\widehat{w}}^{\star }(x)\Vert }^2\forall x\in X_1,\forall w.$

Now let μ = min{δ, c}

${\left(w-{\widehat{w}}^{\star }\right)}^T{\widehat{R}}_{11}(x)(w-{\widehat{w}}^{\star }(x))\le \mu {\Vert (w-{\widehat{w}}^{\star }(x)\Vert }^2\forall x\in X_1.$

Moreover, by hypothesis (ii)

${\widehat{H}}_2({\widehat{w}}^{\star }(x),{\widehat{u}}^{\star }(x))\le -\frac{\mu }{2}({\Vert {\widehat{w}}^{\star }(x)\Vert }^2+{\Vert {\widehat{u}}^{\star }(x)\Vert }^2)\forall x\in X_0.$

Thus, by the triangle inequality,

$\begin{array}{l}{\widehat{H}}_2(w,{\widehat{u}}^{\star }(x))\le -\frac{\mu }{2}(\Vert {\widehat{w}}^{\star }(x){\Vert }^2+\frac{1}{2}\Vert {\widehat{u}}^{\star }(x){\Vert }^2+\frac{1}{2}\Vert (w-{\widehat{w}}^{\star }(x){\Vert }^2)\\ \le -\frac{\mu }{2}\left(\Vert w{\Vert }^2+\Vert {\widehat{u}}^{*}\left(x\right){\Vert }^2\right)\end{array}$

(7.79)

for all x ∈ X₂, where X₂ = X₀ ∩ X₁. Notice however that the Hamiltonians H₂(x, w, u,) and Ĥ₂(x, w, u) defined by (7.72) and (7.74) respectively, are related by

$H_2(x,w,u)={\widehat{H}}_2(x,w,u)+R_m(x,g_1(x)w+g_2(x)u).$

(7.80)

By the result in Section 8.14.3 of reference [94], for all κ > 0, there exist neighborhoods X₃ of x = 0, W₁ of w = 0 and U₁ of u = 0 such that

$\left|R_m(x,g_1(x)w+g_2(x)u)\right|\le \kappa ({\Vert w\Vert }^2+{\Vert u\Vert }^2)\forall (x,w,u)\in X_3\times W_1\times U_1.$

(7.81)

Finally, combining (7.79), (7.80) and (7.81), one obtains an estimate for H₂(w, u (x)), i.e.,

$H_2(w,u^{\star }(x))\le -\frac{\mu }{2}\left(1-\frac{\kappa }{\mu }\right)({\Vert w\Vert }^2+{\Vert {\widehat{u}}^{\star }(x)\Vert }^2)$

Choosing κ < μ and X ⊆ X₃, W₁ ⊆ W, the result follows. □

From the above lemma, one can conclude the following.

Theorem 7.4.1 Consider the discrete-time nonlinear system (7.1), and assume all the hypotheses (i), (ii), (iii) of Lemma 7.4.1 hold. Then the closed-loop system Σda :

${\sum }^{da}:\{\begin{array}{l}{x}_{k+1}=f(x_k)+g_1(x_k)w_k+g_2(x_k)\overline{u}(x_k);x_0=0\\ z_k=h_1(x_k)+k_{11}(x_k)w_k+k_{12}(x_k)\overline{u}(x_k)\end{array}$

(7.82)

with ū(x) = û (x) has a locally asymptotically-stable equilibrium-point at x = 0, and for every K ∈ Z₊, there exists a number > 0 such that the response of the system from the initial state x₀ = 0 satisfies

${\displaystyle \sum _{k=0}^K{\Vert z_k\Vert }^2}\le {\text{γ}}^2{\displaystyle \sum _{k=0}^K{\Vert w_k\Vert }^2}$

for every sequence w = (w₀,…, w_K) such that w_k < ε.

Proof: Since f(.) and û (.) are smooth and vanish at x = 0, it is easily seen that for every K > 0, there exists a number > 0 such that the response of the closed-loop system to any input sequence w = (w₀,…, w_K) from the initial state x₀ = 0 is such that x_k ∈ X for all k ≤ K + 1 as long as w_k < for all k ≤ K. Without any loss of generality, we may assume that is such that w_k ∈ W. In this case, using Lemma 7.4.1, we can deduce that the dissipation-inequality

$V(x_{k+1})-V(x_k)+\frac{1}{2}(z_k^T{z}_k-{\text{γ}}^2|w_k^T{w}_k)\le 0\forall k\le K$

holds. The result now follows from Chapter 3 and a Lyapunov argument. □

Remark 7.4.1 Again, in the case of the linear system Σ^dl (7.17), the result of Theorem 7.4.1 reduces to wellknown necessary and sufficient conditions for the existence of a solution to the linear DSFBNLHICP [89]. Indeed, setting B := [B₁ B₂] and D := [D₁₁ D₁₂], a quadratic function V (x) = ¹₂ x^T P x with P = P ^T > 0 satisfies the hypotheses of the theorem if and only if

$\begin{array}{l}{A}^{T}PA-P+C_1^T{C}_1-F_p^T(R+B^{T}PB)F_p<0\\ D_{11}^T{D}_{11}-{\gamma }^{2}I+B_1^{T}P{B}_1<0\end{array}$

where $F_p=-{\left(R+B^{T}PB\right)}^{-1}(B^{T}PA+D^T{C}_1).$. In this case,

$\left[\begin{array}{c}{\widehat{w}}^{*}\\ {\widehat{u}}^{*}\end{array}\right]=F_{p}x.$

In the next subsection, we consider an approximate approach to the measurement-feedback problem.

7.4.2    An Approximate Approach to the Discrete-Time Output Measurement-Feedback Problem

In this section, we discuss an alternative approximate approach to the discrete-time measurement-feedback problem for affine systems. In this regard, assume similarly a dynamic observer-based controller of the form

${\overline{\sum }}_{dynobs}^{dac}:\{\begin{array}{l}{\theta }_{k+1}=f({\theta }_k)+g_1({\theta }_k){\widehat{w}}^{\star }({\theta }_k)+g_2({\theta }_k){\widehat{u}}^{\star }({\theta }_k)+\overline{G}({\theta }_k)[y_k-h_2({\theta }_k)-\\ k_{21}({\theta }_k){\widehat{w}}^{\star }({\theta }_k)\\ u_k={\widehat{u}}^{\star }({\theta }_k)\end{array}$

(7.83)

where θ ∈ X is the controller state vector, while ŵ (.), û (.) are the optimal state-feedback control and worst-case disturbance given by (7.76) respectively, and $\overline{G}$ is the output-injection gain matrix which is to be determined. Accordingly, the corresponding closed-loop system (7.1), (7.83) can be represented by

$\{\begin{array}{l}{x}_{k+1}^{\#}=f^{\#}(x_k^{\#})+g^{\#}(x_k^{\#})w_k\\ z_k^{\#}=h^{\#}(x_k^{\#})+k^{\#}(x_k^{\#})w_k\end{array}$

(7.84)

where x^# = [x^T θ^T ]^T ,

$f^{\#}(x^{\#})=\left[\begin{array}{l}f(x)+g_2(x){\widehat{u}}^{\star }\left(\theta \right)\\ \left(\begin{array}{l}f(\theta )+g_1\left(\theta \right){\widehat{w}}^{\star }\left(\theta \right)+g_2(\theta ){\widehat{u}}^{\star }\left(\theta \right)+\\ \overline{G}(\theta )(h_2(x)-h_2(\theta )-k_{21}(\theta ){\widehat{w}}^{\star }\left(\theta \right)\end{array}\right)\end{array}\right],$

$g^{\#}(x^{\#})=\left[\begin{array}{l}{g}_1(x)\\ \overline{G}(\theta )k_{21}(x)\end{array}\right],$

and

$h^{\#}(x^{\#})=h_1\left(x\right)+k_{12}(x){\widehat{u}}^{\star }\left(\theta \right),k^{\#}(x^{\#})=k_{11}(x).$

The objective is to find sufficient conditions under which the above closed-loop system (7.84) is locally (globally) asymptotically-stable and the estimate θ → x as t → ∞. This can be achieved by first rendering the closed-loop system dissipative, and then using some suitable additional conditions to conclude asymptotic-stability.

Thus, we look for a suitable positive-definite function Ψ₁ : $\mathcal{X}$ × $\mathcal{X}$ → ₊, such that the dissipation-inequality

${\text{Ψ}}_1(x_{k+1}^{\#})-\text{Ψ}(x_k^{\#})+\frac{1}{2}(|\left|z_k^{\#}\right|{|}^2-{\text{γ}}^2\left|\right|w_k|{|}^2)\le 0$

(7.85)

is satisfied along the trajectories of the system (7.84). To achieve this, we proceed as in the continuous-time case, Chapter 5, and assume the existence of a smooth C² function W : X × X → such that W (x ) ≥ 0 for all x = 0 and W (x ) > 0 for all x = θ. Further, set

$H_2^{\#}(w)=W(f^{\#}(x^{\#})+g^{\#}(x^{\#})w)-W(x^{\#})+H_2(w,{\widehat{u}}^{\star }(\theta ))-{\widehat{H}}_2({\widehat{w}}^{\star }(x){\widehat{u}}^{\star }(x)),$

(7.86)

and recall that

$H_2(w,u)={\widehat{H}}_2(w,u)+R_m(x,g_1(x)w+g_2(x)u);$

so that

$H_2(w,{\widehat{u}}^{\star }(\theta ))={\widehat{H}}_2(w,{\widehat{u}}^{\star }(\theta ))+R_m(x,g_1(x)w+g_2(x){\widehat{u}}^{\star }(x)).$

Moreover, by definition

$\begin{array}{l}{H}_2(w,{\widehat{u}}^{\star }(\theta ))=V(f(x)+g_1(x)w+g_2(x){\widehat{u}}^{\star }(\theta )-V(x)+\frac{1}{2}||h_2(x)+\\ k_{11}(x)w++k_{12}(x){\widehat{u}}^{\star }(\theta )|{|}^2-\frac{1}{2}{\text{γ}}^2|\left|w\right|{|}^2.\end{array}$

(7.87)

Therefore, subsituting (7.87) in (7.86) and rearranging, we get

$\begin{array}{l}{H}_2^{\#}(w)+{\widehat{H}}_2({\widehat{u}}^{\star }(x),{\widehat{w}}^{\star }(x))=W(f^{\#}(x^{\#})+g^{\#}(x^{\#})w)-W(x^{\#})+\\ V(f(x)+g_1(x)w+g_2(x){\widehat{u}}^{\star }(\theta ))-V(x)+\frac{1}{2}\left|\right|h^{\#}(x^{\#})+k^{\#}(x^{\#})w|{|}^2-\frac{1}{2}{\text{γ}}^2|\left|w\right|{|}^2\end{array}$

(7.88)

From the above identity (7.88), it is clear that, if the right-hand-side is nonpositive, then the positive-definite function

${\text{Ψ}}_1(x^{\#})=W(x^{\#})+V(x)$

will indeed have satisfied the dissipation-inequality (7.85) along the trajectories of the closed-loop system (7.84). Moreover, if we assume the hypotheses of Theorem 7.4.1 (respectively Lemma 7.4.1) hold, then the term 5H₂(û (x), ŵ (x)) is nonpositive. Therefore, it remains to impose on H^#₂(w) to be also nonpositive for all w.

One way to achieve the above objective, is to impose the condition that

$\underset{w}{{\displaystyle \max }}{H}_2{}^{{\#}_{}}\left(w\right)<0.$

However, finding a closed-form expression for w^⋆⋆ = arg max{H₂(w)} is in general not possible as observed in the previous section. Thus, we again resort to an approximate but practical approach. Accordingly, we can replace the term W (f (x ) + v ) in H₂(.) by its second-order Taylor approximation as:

$W(f^{\#}(x^{\#})+{\upsilon }^{\#})=W(f^{\#}(x^{\#}))+W_{x\#}(f^{\#}(x^{\#})){\upsilon }^{\#}+\frac{1}{2}{\upsilon }^{{\#}^T}{W}_{x^{\#}{x}^{\#}}(f^{\#}(x^{\#})){\upsilon }^{\#}+R_m^{\#}(x^{\#},{\upsilon }^{\#})$

for any v ∈ $\mathcal{X}$ × $\mathcal{X}$, and where R_m(x , v ) is the remainder term. While the last term (recalling from equation (7.75)) can be represented as

$\begin{array}{l}{H}_2(w,{\widehat{u}}^{\star }(\theta ))-{\widehat{H}}_2({\widehat{w}}^{\star }(x),{\widehat{u}}^{\star }(x))=\frac{1}{2}{\left[\begin{array}{l}w-{\widehat{w}}^{\star }(x)\\ {\widehat{u}}^{\star }(\theta )-{\widehat{u}}^{\star }(x)\end{array}\right]}^T\widehat{R}(x)\left[\begin{array}{l}w-{\widehat{w}}^{\star }(x)\\ {\widehat{u}}^{\star }(\theta )-{\widehat{u}}^{\star }(x)\end{array}\right]\\ +R_m(x,g_1(x)w+g_2(x){\widehat{u}}^{\star }(\theta )).\end{array}$

Similarly, observe that R_m(x, g₁(x)w + g₂(x)û^⋆ (θ)) can be expanded as a function of $x^{\#}$ with respect to w as

$R_m(x,g_1(x)w+g_2(x){\widehat{u}}^{\star }(\theta ))=R_{m0}(x^{\#})+R_{m1}(x^{\#})w+w^T{R}_{m2}(x^{\#})w+R_{m3}(x^{\#},w).$

Thus, we can now approximate H₂(w) with the function

$\begin{array}{l}{H}_2^{\#}(w)=W(f^{\#}(x^{\#}))-W(x^{\#})+R_{m0}(x^{\#})+R_{m1}(x^{\#})+W_{x\#}(f^{\#}(x^{\#}))g^{\#}(x^{\#})w+\\ w^T(R_{m2}(x^{\#})+\frac{1}{2}{g}^{{\#}^T}(x^{\#})W_{x^{\#}{x}^{\#}}(f^{\#}(x^{\#}))g^{\#}(x^{\#}))w+\\ \frac{1}{2}{\left[\begin{array}{l}w-{\widehat{w}}^{\star }(x)\\ {\widehat{u}}^{\star }(\theta )-{\widehat{u}}^{\star }(x)\end{array}\right]}^T\widehat{R}(x)\left[\begin{array}{l}w-{\widehat{w}}^{\star }(x)\\ {\widehat{u}}^{\star }(\theta )-{\widehat{u}}^{\star }(x)\end{array}\right].\end{array}$

(7.89)

Moreover, we can now determine an estimate ŵ of w from the above expression (7.89) for H₂(w) by taking derivatives with respect to w and solving the linear equation

$\frac{\partial {\overline{H}}_2^{\#}}{\partial w}({\widehat{w}}^{\star \star })=0\text{}.$

It can be shown that, if the matrix

$\overline{R}(x^{\#})=\frac{1}{2}{g}^{\#}{\left(x^{\#}\right)}^T{W}_{x^{\#}{x}^{\#}}(f^{\#}(x^{\#}))g^{\#}(x^{\#})+{\widehat{R}}_{11}(x)+R_{m2}(x^{\#})$

is nonsingular and negative-definite, then ŵ is unique, and is a maximum for H₂(w). The design procedure outlined above can now be summarized in the following theorem.

Theorem 7.4.2 Consider the nonlinear discrete-time system (7.1) and suppose the following hold:

(i)  there exists a smooth positive-definite function V defined on a neighborhood X₀ ⊂ $\mathcal{X}$ of x = 0, satisfying the hypotheses of Lemma 7.4.1.

(ii)  there exists a smooth positive-semidefinite function W (x ), defined on a neighborhood Ξ of x = 0 in $\mathcal{X}\times \mathcal{X}$, such that W (x ) > 0 for all x, x = θ, and satisfying

${\overline{H}}_2^{\#}({\widehat{w}}^{\star \star }(x^{\#}))<0$

for all 0 = x ∈ Ξ. Moreover, there exists a number δ > 0 such that

$\begin{array}{l}{\overline{H}}_2^{\#}({\widehat{w}}^{\star \star }(x^{\#}))<-\text{δ||}{\widehat{w}}^{\star \star }(x^{\#})|{|}^2\\ \overline{R}(x^{\#})<0\end{array}$

for all x ∈ Ξ.

Then, the controller${\overline{\Sigma }}_{dynobs}^{dac}$dynobs given by (7.83) locally asymptotically stabilizes the closed-loop system (7.84), and for every K ∈ Z₊, there is a number ε > 0 such that the response from the initial state (x₀, θ₀) = (0, 0) satisfies

${\displaystyle \sum _{k=0}^K{z}_k^T{z}_k\le }{\text{γ}}^2{\displaystyle \sum _{k=0}^K{w}_k^T{w}_k}$

for every sequence w = (w₀,…, w_K) such that w_k² <ε.

7.5    Notes and Bibliography

This chapter is entirely based on the papers by Lin and Byrnes [182]-[185]. In particular, the discussion on controller parameterization is from [184]. The results for stable plants can also be found in [51]. Similarly, the results for sampled-data systems have not been discussed here, but can be found in [124, 213, 255].

The alternative and approximate approach for solving the discrete-time problems is mainly from Reference [126], and approximate approaches for solving the DHJIE can also be found in [125]. An information approach to the discrete-time problem can be found in [150], and connections to risk-sensitive control in [151, 150].